Alternating series are series whose terms alternate in sign between positive and negative. There is a powerful convergence test for alternating series.

*alternating series*). It turns out that there is a powerful test for determining that a series of this form converges.

**alternating series**is a series of the form or of the form

As usual, this definition can be modified to include series whose indexing starts somewhere other than .

**alternating harmonic series**The terms of this series, of course, still approach zero, and their absolute values are monotone decreasing. Because the series is alternating, it turns out that this is enough to guarantee that it converges. This is formalized in the following theorem.

Compared to our convergence tests for series with strictly positive terms, this test is strikingly simple. Let us examine why it might be true by considering the partial sums of the alternating harmonic series. The first few partial sums with odd index are given by

Note that a general odd partial sum is of the form and the quantity in the parentheses is positive. We conclude that:

Moreover, the sequence of odd partial sums is bounded below by zero, since and each quantity in parentheses is positive.

Next consider the sequence

of even partial sums.

Finally, we use and the fact that the limits of the sequences and are finite to conclude that It follows that the alternating harmonic series converges.

With slight modification, the argument given above can be used to prove that the Alternating Series Test holds in general.

The terms of the sequence are positive and nonincreasing, so we can apply the Alternating Series Test. Since the Alternating Series Test implies that the series converges.

Keep in mind that this does not mean we conclude the series diverges; in fact, it does converge. We are just unable to conclude this based on the alternating series test.